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292

Motor Control, 2004, 8, 292-311© 2004 Human Kinetics Publishers, Inc.

293 The Sway-Density Curve

Jacono, Morasso, and Sanguineti are with the Center of Bioengineering, Hospital La Colletta, Arenzano, Italy\aq\, Casadio, Morasso, and Sanguineti are with the Dept. of Informatics, Systems and Telecommunication, University of Genova, Italyn.

The Sway-Density Curve and the Underlying Postural Stabilization Process

Marco Jacono, Maura Casadio,Pietro G. Morasso, and Vittorio Sanguineti

The sway-density curve (SDC) is computed by counting, for each time instant, the number of consecutive samples of the statokinesigram falling inside a circle of “small” radius R. The authors evaluated the sensitivity of the curve to the variation of R and found that in the range 3–5 mm the sensitivity was low, indicating that SDC is a robust descriptor of posturographic patterns. In addition, they investigated the relationship between SDC and the underlying postural stabilization process by decomposing the total ankle torque into three components: a tonic component (over 69% of the total torque), an elastic torque caused by ankle stiffness (about 19%), and an anticipatory active torque (about 12%). The last component, although the smallest in size, is the most critical for the overall stability of the standing posture and appears to be correlated with the SDC curve.

Key Words: ankle stiffness, anticipatory control, postural parameters

Different lines of evidence suggest that the quiet standing posture is stabilized by the brain by means of an anticipatory control mechanism that operates in cooperation with the intrinsic stabilizing action provided by the elastic properties of the ankle muscles.

First of all, if we approximate the body by an inverted pendulum hinged at the ankle, an important empirical finding (Morasso & Schieppati, 1999) is that the control variable center of pressure (COP) and the controlled variable center of mass (COM) are phase locked, and the peak of their cross-correlation function occurs for a zero-time shift. This would not be possible in a feedback-control system in that the delays in the loop (either transmission delays or phase delays resulting from the dynamics of the control modules and the mechanical plant) would inevitably induce the controlled variable to lag the control variable. Such argument is further supported by the electromyographic analysis of the ankle muscles (Gatev et al., 1999) that shows that the activity of ankle muscles is modulated in anticipation (about 250 ms) of the movements of the COM, thus providing the hard evidence for implementing the phase lock mentioned previously.

292

Motor Control, 2004, 8, 292-311© 2004 Human Kinetics Publishers, Inc.

293 The Sway-Density Curve

In any case, elementary control theory shows that the feedback stabilization by means of segmental reflexes is bound to fail because of the substantial delays in the feedback signals: Paradoxically, the reflexes, if present, would increase the intrinsic instability of the plant instead of compensating for it. Supporting this consideration is the fact that the influence of the stretch reflex of the ankle muscles during quiet standing is likely to be minimal because in that condition proprioceptive receptors are stimulated near or below the physiological threshold (Fitzpatrick & McCloskey, 1993; Konradson et al., 1993).

On the other hand, the empirical measurement of ankle stiffness (Casadio et al., 2003; Loram & Lakie, 2002b) has shown that a pure mechanism of stiffness stabilization, advocated by Winter et al. (1998) among others, is not possible because ankle stiffness is smaller than the critical value because of gravity; thus, only an anticipatory control mechanism has the chance to solve the stabilization task by adding small but precisely timed stabilization bursts to the intrinsic stiffness action.

How is this reflected in the usual observables of the posturographic analysis? In a previous article (Baratto et al., 2002) we compared different methods of parameterization of the posturographic data from the point of view of clinical use, and, in particular, we proposed a new method based on the analysis of the sway-density curve (SDC), defined as the time-dependent curve that counts, for each time instant, the number of consecutive samples of the statokinesigram falling inside a circle of “small” radius R. We showed that this method of analysis appears to be more robust and sensitive to pathological conditions than other methods. In the meantime, SDC analysis has been applied with success to clinical studies involving early Parkinson’s patients (Fioretti et al., 2003a), diabetic neuropathy (Fioretti et al., 2003b), and experimental pain (Corbeil et al., 2004). Although the standard posturographic parameters (e.g., sway path and sway area) are empirical descriptors of the posturographic data without any attempt to interpret them from the point of view of biomechanics and motor control, SDC indicators were motivated by qualitative motor-control considerations. The success of the clinical studies is a motivation for reexamining the method from the viewpoint of motor-control issues and for further assessing its robustness, particularly with respect to the role of a key parameter in the SDC analysis: the radius used for the moving window sample count. Moreover, we also intended to pursue further the interpretation of this method of analysis in terms of the underlying motor-control process by showing the correlation between SDC and the control torques.

ExperimentsSeven young adults participated in the experiments. They were asked to stand quietly on a force platform, barefoot, with arms relaxed at their sides. Their feet were abducted at 20°, the heels separated by approximately 2 cm, and their eyes closed. The time window was 40 s (occasionally 90 s was used for illustrative purposes). Background noise was usually low, and trials in which sharp directional sounds unexpectedly occurred were eliminated. Table 1 lists anthropometric data, including the distance h between the ankle and the COM measured using the standard anatomical landmarks. (For the ankle we considered the line that joins the lateral malleoli, and for the COM we assumed it to be located just anterior to the second sacral vertebra, midway between joints.)

294 Jacono et al. 295 The Sway-Density Curve

The sway patterns were acquired by means of a strain-gauge, three-component force platform (Argo model, RGM, Genova, Italy). The resonant frequency of the structure is over 100 Hz. The resolution in the computation of the COP is >0.2 mm.

The trajectory of the COP was sampled at 100 Hz and low-pass filtered with a second-order Butterworth filter with a cutoff frequency of 12.5 Hz. The trajectory was then analyzed with posturographic-analysis software developed in the lab, written in Matlab©, that includes routines for recovering COM from COP, computing SDC and other posturographic parameters, and estimating the control torques.

SDC AnalysisThe displacements between subsequent samples of a statokinesigram (the COP trace on the support surface) are far from uniform, as is apparent in the top portion of Figure 1, showing the evolution over time of the intersample displacements for a statokinesigram sampled at 100 Hz. The bottom portion of Figure 1 shows the histogram of the displacement values. Subsequent displacements tended to be similar, yielding a characteristic wax and wane pattern of samples on the posturographic trace, a pattern that is apparent if we dot the statokinesigram at the sample times (see, for example, the top right portion of Figure 2). In particular, let us compute the median displacement over a posturographic sequence (0.17 mm in the case of Figure 1) and join the fragments in which the intersample displacements are either larger or smaller than the median: The duration of such fragments is of the order of hundreds of milliseconds. This indicates that the underlying mechanism is characterized by a rather slow alternation of two phases: slow phase (corresponding to dense clusters of samples) and quick phase (corresponding to stretched clusters).

Starting from such empirical observations, the SDC was defined as the time-dependent curve that counts, for each time instant, the number of consecutive samples of the statokinesigram that fall inside a circle of “small” radius (typically 2–5 mm; see Figure 2). The sample count is divided by the sampling rate, yielding a time dimension for the ordinate axis. In other words, the SDC is a time-versus-time curve, showing the evolution over time of the stay time of the posturographic trace inside a moving circle of radius R. The curve is filtered with a fourth-order Butterworth filter with a cutoff

Table 1 Participant Data

Participant Age (years) Weight (kg) Height (m) COM distance h (m)

BS 26 65 175 88.8

CA 28 60 170 87.0

JA 32 79 174 88.1

OG 29 58 165 82.4

SQ 22 68 172 86.9

ST 22 61 165 82.4VE 22 61 165 82.4


Figure 1 — Nonuniformity of the statokinesigram. Top: Distance between subsequent samples of the statokinesigram, sampled at 100 Hz. Bottom: Histogram of the intersample distances (each beam is 0.055 mm).


frequency of 2.5 Hz and, as illustrated by the bottom portion of Figure 2, is characterized by a regular sequence of peaks.

The following step is the extraction of the time instants at which the SDC curve reaches a peak, tp(i); i

= 1, 2, … to compute the following triplets of indicator:

Figure 2 — Top: Example of statokinesigram (left) and zoomed fragment (right), dotted at the sampling frequency, with the probe circle centered in two positions. Note that only subsequent and preceding samples of the current sample, contained in the probe circle, contribute to the count used by the sway-density curve (SDC). Bottom: SDC computed over a time window of 10 s, after filtering and peak detection.


• dt(i) = tp(i + 1) – t

p(i), time interval between one peak of the SDC

and the next

• z(i), value of the SDC at the peak time

• |p[tp(i + 1)] – p[t

p (i)]|, distance (on the support surface) between

points of the statokinesigram at consecutive peak times

Finally, such indicators are averaged, yielding the following three SDC parameters:

• MT: mean time interval between successive peaks (in seconds)

• MP: mean value of the peaks (in seconds)

• MD: mean distance between successive peaks (in millimeters)

Influence of the Radius RAn important parameter of the SDC algorithm is the radius R, which must be chosen according to some criterion. Figure 3 shows that with increasing values of R the SDC curve tends to increase, as, by definition, it must. Remarkably, however, the peak sequence appears to be invariant. What does vary with R is the coefficient of variation (the ratio between the standard deviation and the mean), which might be considered a measure of contrast for the SDC curve. In fact, there are two extreme values of R for which the SDC curve is flattened (i.e., the coefficient of variation is reduced to zero): The upper bound is the maximum distance among statokinesigram samples (dependent on the participant); the lower bound is the minimum distance between subsequent samples (dependent on the sampling frequency). For a typical participant and at a sampling frequency of 100 Hz such limit values are, respectively, 35 mm and 0.01 mm.

Figure 3 — Sway-density-curve fragment calculated with different values of R.


Figure 4 — Mean distance between successive peaks (MD) parameter of the sway-density curve (SDC) as a function of R for the 7 participants (S1 to S7).

Figure 5 — Mean time interval between successive peaks (MT) parameter of the sway-density curve (SDC) as a function of R for the 7 participants (S1 to S7).

Between such extremes there is a maximum value for the coefficient of variation; in our participants this was reached in the range 3–5 mm. In any case, the choice of R is not critical because away from the optimal values there is a graceful degradation that is also reflected in mild dependence of the SDC parameters (MD, MT, MP) on R.


Figures 4, 5, and 6 illustrate the degree of sensitivity of these parameters:

• MT is the most stable and has the smallest interindividual variation; this, as explained in the following sections, can be explained with the basic biomechanics of the inverted pendulum.

• MD is little dependent on R but can capture interindividual variations as observed by Baratto et al. (2002).

• MP is sensitive, as well, to interindividual variations but appears to grow with R in an almost linear fashion, at least up to a radius of 5 mm.

Together, these observations emphasize that the choice of R is not critical and that a value between 3 and 5 mm is adequate for most applications. At the same time, the quasilinear dependence of MP on R suggests normalizing such an indicator by substituting it with the slope of the regression line MP versus R. In this way the normalized MP indicator has a dimension of s/mm and represents the stay time inside a circle with a 1-mm radius.

After this sensitivity analysis of the SDC parameters, in the next section we establish a modeling framework for addressing an interpretation of such parameters in terms of motor-control concepts.

Figure 6 — Mean value of the peaks (MP) parameter of the sway-density curve (SDC) as a function of R for the 7 participants (S1 to S7).


Biomechanics and Motor Control

The Biomechanical ModelWe use the inverted-pendulum model for analyzing the biomechanics of sway movements, limiting the analysis to the oscillations in the AP (anteroposterior) plane.

Using the Newton–Euler approach for the derivation of the dynamic equations and with reference to the notation defined in Figure 7, we first write the equilibrium equation of the foot,1

– τa +

mg(u –u

0) = 0 (1)

and then the dynamic equation of the inverted pendulum,2

(2)τ ϑ ϑ ϑ ϑa 0= − = − = + − ⇒ = − −mg u u I mgh I mg y y I mg y u( ) ˙ ˙ ( ) ˙ ( )0

Figure 7 — Scheme of the inverted pendulum. h: distance between the center of mass (COM) and the ankle. ϑ (ankle rotation) and τ a (ankle torque) are positive in the counterclockwise direction. u: position of the center of pressure (COP) on the support surface. y: position of the projection of the COM on the support surface. Both are referred to the ankle position (u0 = y0) and are positive in the forward direction.


The moment of inertia of the inverted pendulum can be written as I = mh2ξ, where ξ is a suitable shape factor (typically 1.32). Therefore, the biomechanics of the body is represented by the following equation:

(3)

By computing the Fourier transform of both members of the equation, it is possible to write the transfer function of a “filter” that allows us to recover the time course of the COM y(t) from the measured curve of the COP u(t):

(4)

Figure 8 shows a COM–COP pair where the COM curve is computed according to Eq. 4. The figure also shows that the estimated curve mildly depends on the parameter ξ when it is varied by ±30% around the nominal value.

The two curves (u and y) represent, respectively, the control and the controlled variables of the sway-stabilization system; they are strongly correlated because it can be shown that the cross-correlation function has a sharp peak (greater than 0.95 in all the participants) for a null time shift. This is not compatible with a feedback controller that would inevitably induce the controlled variable to lag the control variable and can only be explained by a combination of stiffness stabilization and anticipatory compensation.

˙ ( )yg

hy u= −

ξ

Y jU j

g h

g h

( )( )

/

/

ωω

ξω ξ

=+2

Figure 8 — Center of pressure (COP) vs. reconstructed center of mass (COM) for different values of the ξ parameter (± 30% of the nominal value).


To clarify this point, we decompose the total ankle torque τa into two main

components: the anticipatory active component τact

and the restoring elastic component QQQQQQQQwhere K

a is the ankle stiffness and QQ is the reference angle implicitly

selected by the tonic activity of the ankle muscles. We assume that QQ is equal to the average sway angle and QQ is the corresponding COP position3:

(5)

It has been shown that the ankle stiffness is smaller than the critical value mgh (the minimum value that would guarantee a pure stiffness stabilization of the inverted pendulum): K

a = γ · mgh with γ < 1. The upper bound of this range has

been estimated by Loram and Lakie (2002a) to be approximately 0.9, in response to very small perturbations of the order of 0.05°; the lower bound (Casadio et al., 2003) is approximately 0.7 and is associated with larger perturbations of the order of 1°. Introducing this expression in Equation 5 we can compute the following expression for the active component of the ankle torque:

(6)

where QQQQQQQQQQQ is a bias torque that depends on the reference system and does not affect the timing structure of the active torque. Thus, the active torque can be separated into a tonic, bias component and a phasic, zero-mean component. Figure 9 shows a typical fragment of active torque reconstructed according to

ϑ− −Ka( )ϑ ϑ)

ϑy

τ ϑ ϑ τ τ τ ϑ ϑa act elastic act a= − = + = − −I mgh K˙ ( ˆ )

τ ϑ ϑ ϑ ξ γ ϑ

ξξ

γ τ γ τ

act a a a= + − − = − − − − − =

− − − − + = − − +

I K mgh K mhyh

mgh mghy y

hK

mhg

hy u mg y mg y u

˙ ( ) ˆ ˙( )

( ) ( ) ( )

2 0

0 01

)

τ γ0 0= −mg y y( ˆ )

Figure 9 — Reconstructed active torque for different values of γ (.7, .8, .9). The curves are normalized by subtracting the mean over the overall observation time. The bold curve corresponds to the limit case of γ = 1.


Equation 6, after eliminating the bias torque. Positive and negative peaks alternate with a rather regular timing structure: Positive torque peaks correspond to backward stabilization commands and negative peaks to forward stabilization commands. The figure also shows that the profile of the active torque is mildly dependent on the value of γ, and in any case this affects only the amplitude, not the shape and the timing of the curve.

Relationship Between SDC and Active TorqueFigure 10 shows the relationship between the active torque and the SDC, particularly with respect to timing. In fact, the figure shows a close correlation between the peaks of the active torque, irrespective of their sign, and the minimum points of the SDC: “A” is an example of correlation with a positive torque peak and “B” is an example with a negative torque peak. Explaining this in the framework of the control model described above,

• The statokinesigram tends to be stable when the ankle torque is approximately constant, and this corresponds to maximum points in the SDC.

• On the contrary, however, the statokinesigram tends to shift quickly when the ankle torque has strong peaks, and this corresponds to minimum points of the SDC.

The relationship between the two curves can be captured by the cross-correlation function between the SDC and the absolute value of the active torque; in all participants this function has a sharp negative peak for a very small time shift (an

Figure 10 — Relationship between the sway-density curve (SDC) and the reconstructed active torque.


average of less than 10 ms), emphasizing the fact that the SDC and the reconstructed active torque have very similar timing structures and thus are likely to be associated with the same dynamical process. This consideration is further supported by the fact that the average interpeak times for the two curves are quite similar.

Decomposition of the Control ActionAs implied in the preceding, the total ankle torque can be decomposed into three stabilization actions that are illustrated in Figure 11:

τa = τ

o + τ

act + τ

elastic (7)

• The bias torque τ0, that can be attributed to the tonic activity of the

ankle muscles

• The active torque τact

, that can be attributed to the phasic activity of the ankle muscles, allowing the COP to burst back and forth with respect to the underlying COM oscillation

• The elastic torque τelastic

, resulting from the ankle stiffness, that follows the oscillations of the COM

Figure 11 — Decomposition of the total ankle torque into three elements: the bias torque resulting from the tonic muscle activity, the elastic torque resulting from stiffness, and the active torque resulting from the phasic muscle activity. The top thin curve is the total torque, the top bold curve is the sum of the bias and elastic torque, and the bottom curve is the pure elastic torque.


By construction, the active and elastic torques are zero-mean curves. In Table 2, the sizes of the stabilizing torques are listed for each participant, together with the corresponding sizes of the COM–COP difference and of the horizontal component of the ground reaction, which can be computed using the definition f

h = mÿ and applying

Equation 3. The “size” is computed as ± 3SDs around the mean. The elastic and active torques depend on the value of ankle stiffness and, in particular, on the coefficient γ (the fraction of the critical stiffness), which ranges from.7 to .9, as discussed previously. Therefore, Table 2 lists two values, the former corresponding to the lower bound of the range (γ = .7) and the latter to the higher bound (γ = 0.9).

Table 2 Postural Stabilization and Ankle Torques

Participant

Size of the COM–COP

difference (mm)

Size of the horizontal force (N)

Size of the elastic torque

(N · m)

Size of the active torque

(N · m)

Tonic torque (N · m)

BS 3.93 2.12 4.32–5.55 3.75–2.91 12.75CA 2.55 1.30 2.19–2.85 2.19–1.74 11.77

JA 5.39 3.57 6.96–8.97 5.64–4.56 21.49OG 3.06 1.59 4.62–5.91 3.30–2.55 17.07

SQ 2.04 1.18 4.59–5.85 3.31–2.52 20.01ST 2.64 1.44 3.21–4.11 2.34–1.74 17.95VE 2.61 1.42 3.42–4.41 2.46–1.80 17.30

Note. COM = center of mass; COP = center of pressure.

Figure 12 — Total ankle torque vs. sway angle. The samples of the total ankle torque acquired during a typical posturographic acquisition (40 s) are plotted against the corresponding angular values. Two lines are plotted: the regression line of the plot and the line that corresponds to the critical stiffness.


Table 2 illustrates that, as expected, with an increasing value of γ, the size of the elastic force increases and the size of the active torque decreases. In any case, the size of the elastic torque is always greater than the active torque. Together, by taking the average between the two extreme values of γ and averaging for the participant population, total ankle torque is distributed in the following way, as a percentage of the total torque: tonic torque, 69.4 ± 6.1%; elastic torque, 18.8 ± 3.%; and active torque, 11.8 ± 2.6%.

Another way of looking at the postural stabilization process is to plot the total ankle torque versus the sway angle (Figure 12). The result, as observed by Winter et al. (2001), is that the slope of the regression line is higher than the critical level of ankle stiffness given by mgh. For the participants studied here, Table 3 lists the slopes of the regression lines and the corresponding critical stiffness values.

Table 3 Critical Ankle Stiffness Versus Torque-Angle Regression

ParticipantCritical ankle stiffness

(Nm/rad)Slope of the regression line

(Nm/rad)Stability margin

(%)

BS 566.28 633.10 11.80CA 512.08 566.67 10.66JA 683.32 736.07 7.72OG 469.08 506.26 7.93SQ 579.68 660.00 13.86ST 493.34 537.21 8.89VE 493.34 537.43 8.94

On average, the “stability” margin (the fraction of the slope that exceeds the critical value) is 9.9%, quite close to what was found by Winter et al. (2001), who used a shorter time window (20 s instead of 40 s). The authors interpreted the finding as evidence that ankle stiffness is above the critical value, and thus a pure stiffness stabilization of the standing posture is possible. This conclusion was challenged by Morasso and Sanguineti (2002) because it was based on a false assumption: that the descending motor commands did not contribute to the stabilization process. Although the regression is inappropriate for estimating the ankle stiffness, the torque-versus-angle plot is useful for visualizing the interplay of the three stabilizing actions mentioned previously. Figure 13 plots, in the torque-versus-angle plane, the same data shown in Figure 11 as time-dependent signals: The tonic torque is a horizontal line passing through the centroid of the posturographic data, the elastic torque varies along a line whose slope is equal to the ankle stiffness, and the active torque is the remaining part of the total torque (small but crucial for stability).

It should be noted that the assumption of a constant tonic torque must only be considered in the short time, on the order of tens of seconds at most. Slow and smooth modulations of this torque component might implement voluntary shifts of the COP from one part of the support base to another, but this does not affect the basic stabilization mechanism, which relies on the synergy between stiffness and active, phasic torques.


DiscussionA preliminary consideration of this discussion relates to the approximations underlying the modeling and the interpretation of the posturographic data. The main approximation is the inverted pendulum hypothesis, which ignores intersegmental interaction. Other approximations, arising from the main one and from the small size of the sway patterns, allow us to neglect the horizontal component of the ground reaction and to equate the ankle angle and the corresponding sine function. Although we cannot estimate to what extent such approximations affect the estimated torques, we do not think that they can alter the basic framework summarized in the decomposition of the control torques described in the preceding section. There is, however, no doubt that an extension of this study to a multisegment model would be quite useful.

Within the limits of the approximated modeling, but supported by the fact that all evidence indicates that ankle stiffness alone is insufficient to stabilize the upright stance, we direct our attention to the anticipatory active torque that, as shown in the previous sections, only accounts for about 12% of the total ankle torque: As already observed, this control action appears to be characterized by a regular sequence of peaks that are well synchronized with the valleys of the SDC. It has also been shown that the mean time between peaks is rather stable, in the range 0.6–0.7 s. This parameter should be compared with the biomechanical model of the inverted pendulum (see Eqs. 3 and 4), which shows that the time constant of a fall, if uncompensated, is given by the following expression:

Figure 13 — Decomposition of the total ankle torque (depicted in Figure 12 in the torque-vs.-angle plane), as the sum of three components: postural tonic torque (the horizontal line), elastic torque (the line with a slope equal to K

a ), and active torque.


(7)

The ankle stiffness provides an action that tends to counteract the destabilizing torque caused by gravity, and both torques are proportional to the angular sway; thus, the presence of stiffness is equivalent to a reduction of g by a factor 1-γ. In our population, taking for γ the lower value .7, the average value of T is 0.62 s, quite compatible with the interpeak time mentioned previously, and suggests that the interpeak time and the time constant of incipient falls might be the same. More generally, we think that such considerations support the idea that the postural stabilization process operates as a sampled data-control system, synchronized with the sequence of incipient falls; such microfalls are aborted by small anticipatory torque bursts that supplement the restoring torque caused by muscle stiffness. This view is in agreement with what Zatsiorsky and Duarte (2000) call “rambling and trembling in quiet standing” (p. n) and Loram and Lakie (2002a) call “small, ballistic-like, throw and catch movements” (p. n). On the contrary, it disagrees with control models based on PD (proportional and derivative) controllers as in Peterka (2000). In the latter case, the parameters of a PD controller were adapted to fit experimental posturograms; although the fit is fine, in our opinion the reconstructed model is unrealistic in two major respects:

• It requires a proportional gain of the controller (equivalent to ankle stiffness) that is much larger than the critical ankle stiffness, in disagreement with the direct measurements of ankle stiffness mentioned previously (Casadio et al., 2003; Loram & Lakie 2002b).

• It requires a persistent noise source of very large size (about 3 Nm) that is unrealistic, because the best known noise source, the ballistic perturbation caused by the heartbeat, has been estimated to be at least one order of magnitude smaller (Conforto et al., 2001).

In fact, the size of the sway patterns remains a puzzle for postural-control models that rely on overcritical ankle stiffness values. Such models are asymptotically stable and thus cannot explain persistent sway patterns unless suitable “perturbations” are inserted in the model. On the contrary, for models that complement an undercritical ankle stiffness with intermittent stabilization bursts, the residual postural oscillations are an integral part, a “signature,” of the control process. It is not by chance that the size of the noise advocated by Peterka (2000) is quite similar to the size of the active torque estimated previously in this article.

We could also speculate why the relative value of ankle stiffness (γ) has evolved to its current level. A controller based on overcritical stiffness is certainly within the capabilities of the biological hardware, but this would require an increase in the tonic muscle activity of the ankle muscle through a substantial level of coactivation, because it is known that muscle stiffness is approximately proportional to the muscle force, at least for muscle-force levels not approaching the maximum voluntary contraction. By considering the data reported by Weiss et al. (1988), who computed the variation of ankle stiffness as a function of ankle torque, we might estimate that muscle force should almost double only to reach the critical stiffness, with a further increase for

Thg

= ξ


ensuring the known stiffness margin. This would require the ankle muscles to come very close to the maximum voluntary contraction, with a very large value of coactivation. Therefore, although feasible, the stiffness or PD control strategy appears to be quite uneconomical from the energetic point of view, because maintaining a very high level of tonic muscle activity for a long time is certainly fatiguing and undesirable from the biological perspective. Moreover, there is no evidence of a substantial coactivation in the ankle muscles during standing. The strategy suggested in this article, based on a mix of stiffness and active control, is certainly more economical, although much more expensive from the computational point of view.

Nonetheless, if we accept this line of reasoning, why did the brain choose to not utilize the “economical” strategy, by operating with a much lower stiffness level? In fact, if forced, the brain is able to master this kind of strategy—this is what occurs when standing on stilts or when rope walking—because in such situations the ankle stiffness is reduced to zero by definition. The reason that could also explain why standing on stilts is more difficult than standing on feet might be related to Equation 7, which tells us that with a full value of g the average falling time constant for our population is 0.33 s. This is about half the value predicted with an ankle stiffness equal to 70% of the critical stiffness (and only one third for a percentage of 90%). The point is that the computational process, which is supposed to be responsible for generating the active torque, must reconstruct with sufficient precision the current state of the standing body before “shooting” the stabilizing bursts. Three hundred milliseconds might be too few to allow “an internal model of sensorimotor integration” to carry out an optimal state estimation, similar to Kalman filtering, with sufficient accuracy (Wolpert et al., 1995)n. In fact, unconventional standing tasks such as standing on stilts are simplified by artificially increasing the height or the moment of inertia: In both cases the result is a decrease of the falling time constant. Moreover, the Wolpert study, although aimed at a different sensorimotor task, provides empirical evidence that Kalman-like sensorimotor filtering processes are commonly employed, and their timing cycles impose specific constraints on the ability to carry out critical integration tasks, particularly in unstable situations.

The same framework could explain why people choose an average standing posture that is displaced slightly forward with respect to the ankle joint (typically 2–3 cm). The common argument that this increases the “static” stability is quite weak because, on one hand, the size of the postural sway is compatible with a reference posture shifted backward onto the position of the ankle joint and, on the other, a strategy of this type should aim at the best possible posture, the midpoint of the foot plant. The alternative explanation is that, at the autonomously chosen angle of forward tilt, the tonic activity of the plantar flexors determined by gravity sets the muscles in an operating point such that the muscle stiffness is just what is needed for achieving the correct amount of slowing of the incipient falls. Thus, it is not the improved biomechanical stability per se that would matter but the gained time for the internal sensorimotor integration process. This type of reasoning also applies to the analysis of pathological conditions, as in the case of Parkinson’s disease, that are characterized, among other things, by an increase of rigidity and thus of stiffness. It might be asked, if this is the case, why does the brain choose to not increase the forward tilt until the critical stiffness is reached? The answer is that, in addition to being undesirable, this is impossible if we consider the stiffness-versus-bias torque curve estimated by Hunter and Kearney (1982) in


relation to the maximum forward tilt allowed by the foot size—critical stiffness would be reached beyond the foot length.

What predictions can be derived from the proposed modeling framework? For example, in rehabilitating patients with neurological deficits, we might consider the basic idea of decomposing the stabilizing control action into a biomechanical part (caused by ankle stiffness) and a cybernetic part (caused by anticipatory commands) a very useful guideline in monitoring functional recovery. In fact, we could expect that in the course of rehabilitation the former stabilization action is decreased in favor of the latter. Moreover, a detailed analysis of the timing of the active control might provide insights into the reorganization of the control action, as is also revealed by the associated SDC parameters.

AcknowledgmentsThis work was partly supported by MIUR [the Italian Ministry of Instruction, Universities, and Research] (COFIN2001 and FIRB2003) and AISM [the Multiple Sclerosis Association of Italy].

Notes1In this equation we neglect the contributions resulting from, respectively, the horizontal component of the ground reaction and the weight of the foot. 2The following approximations are assumed:QQQQQQQQQQQQQQQQQQQQQ; we also assume that u

0 = y

0.

3With the usual small-angle approximation QQQQQQQQQQ, where y0 is the

position of the ankle on the support surface (see Figure 7).

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y y h h y h− = − ( ) ≈ − ≈ −0 sin ϑ ϑ ϑ; ˙ ˙ˆ ˆ /ϑ ≈ − −( )y y h0


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